The Aumann bit is him quoting something which doesn't actually prove what he's quoting it to prove, but which he knows his opponent can't refute because he's never heard of it. It isn't him throwing his hands up in the air--it's an argument, just a fallacious one.
Both times he used it, he's giving up on getting somewhere and is just screwing with the guy; it's not part of his main argument.
The first time he's trying to stop him from weaseling out. Plus, Aumann doesn't mean that, taken in its literal form. But, it applies indirectly, aspirationally: to try to be rational and try to share relevant information, etc. so as to approximate the conditions under which it would apply. Indeed, the most reasonable interpretation of the other's suggestion to agree to disagree is that they both stop trying to be more right than...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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